Metric tensor

The metric tensor's elements are the coefficients read off of the line element For special relativity rectilinear coordinate inertial frames are used which given the metric tensor will be designated $$\eta _{\mu \nu}$$ and the line element will be and the Minkowski metric tensor elements given by

$$\eta _{00} = 1$$

$$\eta _{11} = \eta _{22} = \eta _{33} = -1$$

All other elements are 0.

Written as a matrix this is

The metric tensor acts a an index raising and lowering opperator. And as an inner product operator in 4d spacetime There is an inverse relationship between the contravariant and covariant metric tensor elements which can be expressed as the matrix The covariant derivative of the metric with respect to any coordinate is zero

where the covariant derivative is done with the use of Christoffel symbols. And so of course the covariant divergence of the metric is also zero